How do the typical L^q-dimensions of measures behave? F. Bayart 28A78multifractal analysismeasuresdimensions We compute, for a compact set $K\subset\mathbb{R}^d$, the value of the upper and of the lower $L^q$-dimension of a typical probability measure with support contained in $K$, for any $q\in\mathbb{R}$. Different definitions of the "dimension" of $K$ are involved in computing these values, following $q\in\mathbb{R}$. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5259 10.1512/iumj.2014.63.5259 en Indiana Univ. Math. J. 63 (2014) 687 - 726 state-of-the-art mathematics http://iumj.org/access/